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Atypical late-time singular regimes accurately diagnosed in stagnation-point-type solutions of 3D Euler flows

Published online by Cambridge University Press:  11 January 2016

Rachel M. Mulungye
Affiliation:
Complex and Adaptive Systems Laboratory, School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland
Dan Lucas*
Affiliation:
DAMTP, University of Cambridge, Cambridge CB3 0WA, UK
Miguel D. Bustamante
Affiliation:
Complex and Adaptive Systems Laboratory, School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland
*
Email address for correspondence: dl549@cam.ac.uk

Abstract

We revisit, both numerically and analytically, the finite-time blowup of the infinite-energy solution of 3D Euler equations of stagnation-point type introduced by Gibbon et al. (Physica D, vol. 132, 1999, pp. 497–510). By employing the method of mapping to regular systems, presented by Bustamante (Physica D, vol. 240 (13), 2011, pp. 1092–1099) and extended to the symmetry-plane case by Mulungye et al. (J. Fluid Mech., vol. 771, 2015, pp. 468–502), we establish a curious property of this solution that was not observed in early studies: before but near singularity time, the blowup goes from a fast transient to a slower regime that is well resolved spectrally, even at mid-resolutions of $512^{2}.$ This late-time regime has an atypical spectrum: it is Gaussian rather than exponential in the wavenumbers. The analyticity-strip width decays to zero in a finite time, albeit so slowly that it remains well above the collocation-point scale for all simulation times $t<T^{\ast }-10^{-9000}$, where $T^{\ast }$ is the singularity time. Reaching such a proximity to singularity time is not possible in the original temporal variable, because floating-point double precision (${\approx}10^{-16}$) creates a ‘machine-epsilon’ barrier. Due to this limitation on the original independent variable, the mapped variables now provide an improved assessment of the relevant blowup quantities, crucially with acceptable accuracy at an unprecedented closeness to the singularity time: $T^{\ast }-t\approx 10^{-140}$.

Type
Rapids
Copyright
© 2016 Cambridge University Press 

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References

Bardos, C. & Titi, E. 2007 Euler equations for incompressible ideal fluids. Russ. Math. Surveys 62 (3), 409451.Google Scholar
Beale, J., Kato, T. & Majda, A. 1984 Remarks on the breakdown of smooth solutions for the 3D Euler equations. Commun. Math. Phys. 94, 6166.Google Scholar
Bustamante, M. D. 2011 3D Euler equations and ideal MHD mapped to regular systems: probing the finite-time blowup hypothesis. Physica D 240 (13), 10921099.Google Scholar
Bustamante, M. D. & Brachet, M. 2012 Interplay between the Beale–Kato–Majda theorem and the analyticity-strip method to investigate numerically the incompressible Euler singularity problem. Phys. Rev. E 86 (6), 066302.Google Scholar
Constantin, P. 2000 The Euler equations and nonlocal conservative Riccati equations. Intl Math. Res. Not. 2000 (9), 455465.Google Scholar
Fefferman, C. L. 2000 Existence and smoothness of the Navier–Stokes equation. In The Millennium Prize Problems, pp. 5767. Clay Mathematics Institute.Google Scholar
Gibbon, J. 2008 The three-dimensional Euler equations: where do we stand? Physica D 237, 18941904.Google Scholar
Gibbon, J. D., Fokas, A. S. & Doering, C. R. 1999 Dynamically stretched vortices as solutions of the 3D Navier–Stokes equations. Physica D 132, 497510.CrossRefGoogle Scholar
Gibbon, J. D. & Ohkitani, K. 2001 Singularity formation in a class of stretched solutions of the equations for ideal magneto-hydrodynamics. Nonlinearity 14 (5), 12391264.CrossRefGoogle Scholar
Hou, T. Y. & Li, R. 2006 Dynamic depletion of vortex stretching and non-blowup of the 3D incompressible Euler equations. J. Nonlinear Sci. 16 (6), 639664.Google Scholar
Mulungye, R. M., Lucas, D. & Bustamante, M. D. 2015 Symmetry-plane model of 3D Euler flows and mapping to regular systems to improve blowup assessment using numerical and analytical solutions. J. Fluid Mech. 771, 468502.CrossRefGoogle Scholar
Ohkitani, K. & Gibbon, J. D. 2000 Numerical study of singularity formation in a class of Euler and Navier–Stokes flows. Phys. Fluids 12, 31813194.CrossRefGoogle Scholar
Perlin, M. & Bustamante, M. D. 2015 A robust quantitative comparison criterion of two signals based on the Sobolev norm of their difference. J. Engng Math. (in press).Google Scholar